Abstract
In this paper we present a class of 2D skew-cyclic codes over \(R={\mathbb {F}}_{q}+u{\mathbb {F}}_{q}, u^2=1\), using the bivariate skew polynomial ring \(R[x,y,\theta ,\sigma ]\), where \({\mathbb {F}}_q\) is a finite field, and \(\theta \) and \(\sigma \) are two commuting automorphisms of R. After defining a partial order on \(R[x,y,\theta ,\sigma ],\) we obtain division algorithm for \(R[x,y,\theta ,\sigma ]\) under two different conditions. The structure of 2D skew-cyclic codes over R is obtained in terms of their generating sets. For this, we have classified these codes into different classes, based on certain conditions they satisfy, and accordingly obtained their generating sets in each case separately. A decomposition of a 2D skew-cyclic code C over R into 2D skew-cyclic codes over \({\mathbb {F}}_{q}\) is studied and some examples are given to illustrate the results.
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References
Abualrub, T., Siap, I., Aydin, N.: \({\mathbb{Z}}_2{\mathbb{Z}}_4\)-additive cyclic codes. IEEE Trans. Inf. Theory 60, 1508–1514 (2014)
Abualrub, T., Siap, I., Aydogdu, I.: \({\mathbb{Z}}_2 ({\mathbb{Z}}_2 + u{\mathbb{Z}}_2)\)- linear cyclic codes. In: International MultiConference of Engineers and Computer Scientists (IMECS’2014), vol. II, Hong Kong (2014)
Aydogdu, I., Siap, I.: On \({\mathbb{Z}}_{p^r}{\mathbb{Z}}_{p^s}\)-additive codes. Linear Multilinear Algebra 63, 2089–2102 (2015)
Boucher, D., Ulmer, F.: Coding with skew polynomial rings. J. Symb. Comput. 44, 1644–1656 (2009)
Boucher, D., Geiselmann, W., Ulmer, F.: Skew cyclic codes. Appl. Algebra Eng. Commun. Comput. 18, 379–389 (2007)
Boucher, D., Ulmer, F.: Codes as modules over skew polynomial rings. In: Proceedings of 12th IMA International Conference, Cryptography and Coding, Cirencester, UK, LNCS 5921, pp. 38–55 (2009)
Boucher, D., Solé, P., Ulmer, F.: Skew constacyclic codes over Galois rings. Adv. Math. Commun. 2, 273–292 (2008)
Ikai, T., Kosako, H., Kojima, Y.: Two-dimensional cyclic codes. Electron. Commun. Jpn. 57A, 27–35 (1975)
Imai, H.: A theory of two-dimensional cyclic codes. Inf. Control 34, 1–21 (1977)
Ribenboim, P.: Sur la localisation des anneaux non commutatifs, Seminaire Dubreil. Algebre et theorie des nombres, tome 24 (1970–1971)
Sharma, A., Bhaintwal, M.: \(F_3R\)-skew cyclic codes. Int. J. Inf. Coding Theory 3(3), 234–251 (2016)
Xiuli, L., Hongyan, L.: 2-D skew cyclic codes over \({\mathbb{F}}_{q}[x, y;\rho,\theta ]\). Finite Fields Appl. 25, 49–63 (2014)
Yildiz, B., Aydin, N.: On cyclic codes over \({\mathbb{Z}}_4 + u{\mathbb{Z}}_4\) and their \({\mathbb{Z}}_4\)-images. Int. J. Inf. Coding Theory 2, 226–237 (2014)
Acknowledgements
This work was partially supported by DST, Govt. of India, under Grant No. SB/S4/MS: 893/14. Also, the first author would like to thank the Council of Scientific & Industrial Research (CSIR), India for providing financial support.
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Sharma, A., Bhaintwal, M. A class of 2D skew-cyclic codes over \({\mathbb {F}}_{q}+u{\mathbb {F}}_{q}\). AAECC 30, 471–490 (2019). https://doi.org/10.1007/s00200-019-00388-w
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DOI: https://doi.org/10.1007/s00200-019-00388-w